4 INTRODUCTION

by Bonfiglioli, Lanconelli, Uguzzoni in [3], [4], [5] that the fundamental solution h

exists and satisfies Gaussian bounds of the kind (0.4). As a consequence of these

estimates, in [6] it is proved a scaling invariant Harnack inequality for the operator

H.

A particular class of operators of the kind (0.6), namely ultraparabolic opera-

tors of Kolmogorov-Fokker-Planck type, has been studied by Pascucci and Polidoro

in relation both with Harnack inequality and Gaussian bounds for the fundamental

solution; see [47].

Previous results about Harnack inequality for general H¨ ormander’s operators

date back to Bony’s seminal paper [8], where a first qualitative version of this result

is proved. A first scaling invariant Harnack inequality for heat-type H¨ormander’s

operators was proved later by Kusuoka-Stroock [35].

Strategy and structure of the paper. Following the general strategy used

in the case of homogeneous groups in [3], [4], [6], our study proceeds in three steps,

corresponding to the Parts of this paper. In Part I we consider operators of kind

(0.1) with constant coeﬃcients aij, and no lower order terms. For these operators,

existence and basic properties of the fundamental solution hA are guaranteed by

known results (see Section 3). Here the point is to prove sharp Gaussian bounds

on hA, which have to be uniform in the ellipticity class of the matrix A = {aij} .

In Part II we study operators with variable H¨ older continuous coeﬃcients

aij,ak,a0, and apply the results of Part I to establish existence and Gaussian bounds

for the fundamental solution of these operators. This is accomplished by a suitable

adaptation to our subelliptic context of the classical Levi’s parametrix method.

Finally, thanks to the results of Part II, the proof of a Harnack inequality for

H can follow the lines drawn in [6], and inspired by Fabes-Stroock’s paper [22]:

this is accomplished in Part III.

For the reader’s convenience, we have included at the beginning of each Part

of the paper more details about the strategy, the techniques, and the main new

diﬃculties we had to overcome to reach our results.

A motivation. Many problems in geometric theory of several complex vari-

ables lead to fully nonlinear second order equations, whose linearizations are non-

variational operators of H¨ ormander type (0.5). Here we would like to present one

of these problems whose source goes back to some papers by Bedford, Gaveau,

Slodkowsky and Tomassini, see [2], [56], [54].

Let M be a real hypersurface, embedded in the Euclidean complex space

Cn+1.

The Levi form of M at a point p ∈ M is a Hermitian form on the complex tan-

gent space whose eigenvalues λ1 (p) , ..., λn (p) determine in the directions of each

corresponding eigenvector a kind of “principal curvature”. Then, given a general-

ized symmetric function s, in the sense of Caffarelli-Nirenberg-Spruck [14], one can

define the s-Levi curvature of M at p, as follows:

Sp (M) = s (λ1 (p) , ..., λn (p)) .

When M is the graph of a function u and one imposes that its s-Levi curvature

is equal to a given function, one obtains a second order fully nonlinear partial

differential equation, which can be seen as the pseudoconvex counterpart of the

usual fully nonlinear elliptic equations of Hessian type, as studied e.g. in [14]. In

linearized form, the equations of this new class can be written as (see [43, equation